Introductory Logic Test #3 April 19, 2006 R. Hammack Name: ________________________ Score: _________

1. Use only the 18 rules of implication or replacement to derive the conclusions of the following arguments.

 (a) 1.   M ⊃ S 2.    K ∨ ~S 3.    ~K 4.    ~M ⊃ P /   P 5.    ~S 2, 3, DS 6.    ~M 1, 5, MT 7.    P 4, 7, MP
 (b) 1.   A ∨ B 2.   A ⊃ B /    B 3.   ~~A ∨ B 1, DN 4.   ~A ⊃ B 3, Impl. 5.   ~B ⊃ ~A 2, Trans 6.    ~B ⊃ B 5, 4, HS 7.   ~~B ∨ B 6, Impl. 8.   B ∨ B 7, DN 9.   B 8, Taut.

 (c) 1.   A ⊃ ~(~X • Y) 2.   (X ∨ ~Y) ⊃ Y /    A ⊃ Y 3.   A ⊃ (~~X ∨ ~Y) 1, DM 4.   A ⊃ (X ∨ ~Y) 3, DN 5.   A ⊃ Y 4, 2, HS

 (d) 1.    D ⊃ B 2.    C ≡ D 3.    C /   D • B 4.    (C ⊃ D) • (D ⊃ C) 2, Equiv. 5.    C ⊃ D 4, Simp. 6.    D 5, 3, MP 7.    B 1, 6, MP 8.    D • B 6, 7, Conj
 (e) 1.   L ∨ (M • G) 2.    ~M /   L 3.    (L ∨ M) • (L ∨ B) 1, Dist. 4.    L ∨ M 3, Simp 5.    M ∨ L 4, Comm. 6.    L 5, 2, DS
 (f) If sports shoe manufacturers decline to use kangaroo hides in their products, then Australian hunters will cease killing millions of kangaroos yearly. It is not the case that both Australian hunters will cease killing millions of kangaroos yearly and the kangaroo not be saved from extinction. Therefore, if sports shoe manufacturers decline to use kangaroo hides in their products, then the kangaroo will be saved from extinction.
 1.   S ⊃ H 2.   ~(H • ~E) /   S ⊃ E 3.   ~H ∨ ~~E 2, DM 4.    ~H ∨ E 3, DN 5.   H ⊃ E 4, Impl 6.   S ⊃ E 1, 5 HS

2. Use the technique of conditional proof to deduce the conclusion of the following argument.

 1.   (G ∨ A) ⊃ (S • V) 2.   V ⊃ (C • D) /     G ⊃ D 3.   G ACP 4.   G ∨ A 3, Add 5.   S • V 1, 4, MP 6.   V 5, Comm, Simp 7.   C • D 2, 4, MP 8.   D 7, Comm, Simp 9.   G ⊃ D 3-7, CP

3.  Use the technique of indirect proof to deduce the conclusion of the following argument.

 1.   (R ∨ Q) ⊃ K 2.   ~R ⊃ ~P 3.   ~Q ⊃ P /    K 4.   ~K ACP 5.   ~(R ∨ Q) 1, 4, MT 6.   ~R • ~Q 5, DM 7.   ~R 5, Simp 8.   ~Q 5, Comm, Simp 9.   ~P 2, 7, MP 10.   P 3, 8, MP 11.   P • ~P 9, 10 Conj 12.   K 4-11, IP

4. Use any method from Chapter 7 to deduce the conclusions of the following arguments.
 (a) 1.   L ⊃ (~C ⊃ N) 2.   ~N • P /    L ⊃ (C • P) 3.   L ACP 4.   ~C ⊃ N 1, 3, MP 5.   ~~C ∨ N 4, Impl 6.   C ∨ N 5, DN 7.   ~N 2, Simp 8.   C 6, 7, Comm, DS 9.   P 2, Comm, Simp 10.   C • P 8, 9, Conj 11.  L ⊃ (C • P) 3-10, CP

 (b) 1.   (A ∨ B) ⊃ (D • C) 2.   C ⊃ ~D /   ~A 3.   ~~A AIP 4.   A 3, DN 5.   A ∨ B 5, Add 6.   D • C 1, 5, MP 7.   C 6, Comm, Simp 8.   D 6, Simp 9.   ~D 2, 7, MP 10.   D • ~D 8, 9 Conj 11.   ~A 3-10, IP